Interquartile Range vs Standard Deviation: When to Use Each

Quick Answer

Interquartile range (IQR) measures the spread of the middle 50% of values. Standard deviation measures typical distance from the mean using every value. Use IQR when the median, quartiles, skew, or outliers matter most. Use standard deviation when the mean is meaningful and the next step involves z-scores, confidence intervals, control limits, or other variance-based methods.

Rule of thumb

Report IQR with the median. Report standard deviation with the mean. If you report mean plus IQR or median plus standard deviation, make sure you have a specific reason.

For the broader landscape of spread measures, start with What Is Standard Deviation?, then compare this article with Standard Deviation vs Range and Robust Statistics: MAD and IQR.

What Each Measure Captures

IQR focuses on the center of the distribution. It ignores the lowest 25% and highest 25% when calculating spread, so it is resistant to extreme values. This makes it useful for salaries, home prices, response times, lab measurements with occasional contamination, and any distribution where the middle case is more representative than the average.

Standard deviation uses every observation and measures spread around the mean. Because deviations are squared before they are averaged, large deviations receive extra weight. That sensitivity is a feature when extreme values are part of the real process, but it is a liability when a few errors or unusual cases should not define typical variation.

Question	IQR answers	Standard deviation answers
What center does it pair with?	Median	Mean
Which values drive it?	Q1 and Q3, the quartile boundaries	Every value in the dataset
How does it react to outliers?	Usually changes little unless outliers affect quartiles	Can change substantially because large deviations are squared
What does it describe?	Width of the middle half of the data	Typical spread around the mean
Common use cases	Box plots, skewed data, robust summaries, outlier fences	Normal models, z-scores, confidence intervals, process variation

Formulas and Units

Interquartile Range

IQR = Q3 - Q1

Sample Standard Deviation

s = sqrt(sum((x_i - x_bar)^2) / (n - 1))

Both measures stay in the original units. If the data are in dollars, IQR and standard deviation are in dollars. If the data are in milliseconds, both are in milliseconds. The difference is not units, but which part of the distribution each measure emphasizes.

Normal-distribution conversion

For approximately normal data, IQR is about 1.349 standard deviations. A rough conversion is SD estimate = IQR / 1.349. Do not use this shortcut for strongly skewed or multi-peaked data.

Worked Example

Consider delivery times in minutes: `18, 19, 20, 20, 21, 22, 23, 24, 25`. The median is `21`. Using the lower half `18, 19, 20, 20`, Q1 is `(19 + 20) / 2 = 19.5`. Using the upper half `22, 23, 24, 25`, Q3 is `(23 + 24) / 2 = 23.5`. So IQR is `23.5 - 19.5 = 4` minutes.

The mean is about `21.33` minutes, and the sample standard deviation is about `2.35` minutes. In this clean, roughly symmetric dataset, both measures tell a compatible story: most delivery times are close together.

Dataset	Median	IQR	Mean	Sample SD
18, 19, 20, 20, 21, 22, 23, 24, 25	21.00	4.00	21.33	2.35

Practical reading

A manager could say the middle half of deliveries spans about 4 minutes, while the typical distance from the mean is about 2.35 minutes. IQR gives a percentile-based summary. Standard deviation gives the spread measure needed for z-score interpretation or confidence intervals.

Outlier Effect

Now replace the last delivery time with a severe delay: `18, 19, 20, 20, 21, 22, 23, 24, 60`. The median stays `21`. Q1 remains `19.5`, Q3 remains `23.5`, and IQR remains `4`. The mean rises to about `25.22`, and the sample standard deviation jumps to about `13.13`.

Dataset	Median	IQR	Mean	Sample SD	Interpretation
Clean delivery times	21.00	4.00	21.33	2.35	Middle and mean-based spread agree
One severe delay	21.00	4.00	25.22	13.13	SD highlights the severe tail event

Neither result is automatically wrong

IQR is telling you the typical middle of the process did not change. Standard deviation is telling you the process now has a major tail-risk event. The right measure depends on the question.

When to Use IQR

Use IQR for skewed distributions such as income, wait times, order values, home prices, and time-to-resolution metrics.
Use IQR when the median is the main center measure and you want a matching spread measure.
Use IQR for box plots and percentile summaries.
Use IQR when you need the 1.5 x IQR outlier rule. For a full workflow, see Detecting Outliers with Standard Deviation.
Use IQR when occasional data-entry mistakes or rare events should not dominate the headline measure.

When to Use Standard Deviation

Use standard deviation when data are roughly symmetric and the mean is a meaningful center.
Use standard deviation for z-scores, normal models, standard error, confidence intervals, and hypothesis tests.
Use standard deviation when every observation should contribute to the measure of variation.
Use standard deviation for quality control and process monitoring where unusual variation should be visible.
Use standard deviation when you need compatibility with formulas covered in Standard Deviation Formula Explained.

Use the percentile calculator for IQR

The percentile calculator helps find Q1, Q3, and IQR from raw values, which is the fastest way to build a median-plus-IQR summary.

Use the standard deviation calculator for SD

The standard deviation calculator computes sample SD, population SD, variance, mean, range, and step-by-step work from the same raw dataset.

Decision Checklist

Center:If you report the median, pair it with IQR. If you report the mean, pair it with standard deviation.
Shape:If the distribution is skewed, heavy-tailed, or has clear outliers, IQR is usually the more stable summary.
Purpose:If the next calculation needs variance, standard error, z-scores, or normal assumptions, standard deviation is usually required.
Audience:For nontechnical percentile summaries, IQR is often easier to explain. For statistical reports, standard deviation is usually expected.
Risk:If rare extreme values are operationally important, do not hide them behind IQR alone. Report IQR plus a tail metric such as max, range, or percentiles.

A strong report often includes both: `median (IQR)` to describe the typical middle and `mean (SD)` to support statistical modeling. When the two tell different stories, that difference is useful evidence about skew, outliers, or tail risk.

Sources

References and further authoritative reading used in preparing this article.

Interquartile range — Wikipedia
Standard deviation — Wikipedia
NIST/SEMATECH e-Handbook of Statistical Methods — NIST

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